Cophenetic correlation

In statistics, and especially in biostatistics, cophenetic correlation^[1] (more precisely, the cophenetic correlation coefficient) is a measure of how faithfully a dendrogram preserves the pairwise distances between the original unmodeled data points. Although it has been most widely applied in the field of biostatistics (typically to assess cluster-based models of DNA sequences, or other taxonomic models), it can also be used in other fields of inquiry where raw data tend to occur in clumps, or clusters.^[2] This coefficient has also been proposed for use as a test for nested clusters.^[3]

Calculating the cophenetic correlation coefficient

Suppose that the original data {X_i} have been modeled using a cluster method to produce a dendrogram {T_i}; that is, a simplified model in which data that are "close" have been grouped into a hierarchical tree. Define the following distance measures.

• x(i, j) = | X_i − X_j |, the ordinary Euclidean distance between the ith and jth observations.
• t(i, j) = the dendrogrammatic distance between the model points T_i and T_j. This distance is the height of the node at which these two points are first joined together.

Then, letting x be the average of the x(i, j), and letting t be the average of the t(i, j), the cophenetic correlation coefficient c is given by^[4]

$c = \frac {\sum_{i<j} (x(i,j) - x)(t(i,j) - t)}{\sqrt{[\sum_{i<j}(x(i,j)-x)^2] [\sum_{i<j}(t(i,j)-t)^2]}}.$

See also

• Cophenetic

References

1. ^ Sokal, R. R. and F. J. Rohlf. 1962. The comparison of dendrograms by objective methods. Taxon, 11:33-40
2. ^ Dorthe B. Carr, Chris J. Young, Richard C. Aster, and Xioabing Zhang, Cluster Analysis for CTBT Seismic Event Monitoring (a study prepared for the U.S. Department of Energy)
3. ^ Rohlf, F. J. and David L. Fisher. 1968. Test for hierarchical structure in random data sets. Systematic Zool., 17:407-412
4. ^ Mathworks statistics toolbox

External links

• Numerical example of cophenetic correlation